(3/19/08)
CHAPTER 2: THE DERIVATIVE AND APPLICATIONS
The ancient Greeks did some amazing mathematics. Their work on the theory of proportions, in plane
geometry, and on tangent lines, areas, and volumes came close to meeting modern standards of exposition
and logic. Their explanations of topics from physics, such as the motion of projectiles and falling bodies, in
contrast, were very different from modern theories. The philosopher Aristotle (384–322 BC), for example,
said that the motion of projectiles is caused by air pushing them from behind and that bodies fall with
speeds proportional to their weights.
(1)
His point of view was generally not questioned in Western Europe
until the Renaissance, when calculus was developed to study rates of change of nonlinear functions
that arise in the study of motion and Newton explained how forces such as air resistance and gravity
affect motion. We see in this chapter that the rate of change of a function is its
derivative
, which
is the slope of a tangent line to its graph. We study linear functions and constant rates of change in
Section 2.1 and average rates of change in Section 2.2. In Section 2.3 we give the general definition of the
derivative as a limit of average rates of change. Formulas for exact derivatives of powers of
x
and of linear
combinations of functions are derived in Section 2.4. In Section 2.5 we look at derivatives as functions
and discuss applications that require finding approximate derivatives from graphs and tables. Rules for
finding derivatives of products, quotients, and powers of functions are discussed in Sections 2.6 and 2.7.
Section 2.8 deals with linear approximations and differentials.
Section 2.1
Linear functions and constant rates of change
Overview:
As we will see in later sections, the rates of change of most functions are found by using
calculus techniques to find their derivatives. In this section, however, we study applications that involve
a special class of functions whose rates of change can be determined without calculus. These are the
linear functions
. A linear function is a firstdegree polynomial
y
=
mx
=
b
. Its graph is a line and its
(constant) rate of change is the slope of its graph.
Topics:
•
Constant velocity
•
Other constant rates of change
•
Approximating data with linear functions
Constant velocity
If an object moves on a straight path, we can use an
s
axis along that path, as in Figure 1, to indicate
the object’s position. The object’s
s
coordinate is then a function of the time
t
, which is linear if the
object’s velocity is constant.
s

100
0
100
200
300
400
500
FIGURE 1
Example 1
A moving van is 200 miles east of a city at noon and is driving east at the constant
velocity of 60 miles per hour. Give a formula for the van’s distance
s
=
s
(
t
) east of the
city
t
hours after noon.
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 Summer '08
 Eggers
 Math, Geometry, Derivative, tc, Linear function, Eb

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